Endomorphism/Basis/To scalar multiples/Dimension/Exercise

Let ${}V$ be a ${}K$-vector space, and let ${}v_{1},\ldots ,v_{n}$ denote a basis of ${}V$. Determine the dimension of the space of all endomorphisms

$\varphi \colon V\longrightarrow V$

satisfying

${}\varphi (v_{i})\in Kv_{i}\,$

for all ${}i$. How do the matrices of such a ${}\varphi$ with respect to this basis look like?

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